"I am sad that a terrorist bomb attack just killed 20 people in Sabon Gari Zaria," Nasir Ahmad El-Rufai, the governor of Kaduna state, wrote on his Twitter account.
The blast happened at about 9:00 am (0800 GMT) as primary school teachers and public sector staff underwent screening at government offices in the city, according to witnesses.
There was no immediate claim of responsibility but Boko Haram militants will be the main suspects, as they have carried out a wave of similar attacks in the past.
El-Rufai, from the All Progressives Congress (APC) party of President Muhammadu Buhari, said people in the mainly Muslim city should be vigilant and avoid crowded places in the next few weeks, AFP reported.
"We call on our people to report any suspicious persons and movements to law enforcement agents, traditional rulers and religious leaders," he added.
"The Kaduna State government is working with Federal security agencies that are taking proactive measures to frustrate such terrorists."